Biostat/Biomath M257
System information (for reproducibility):
versioninfo()Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: macOS (arm64-apple-darwin24.0.0)
CPU: 12 × Apple M2 Max
WORD_SIZE: 64
LLVM: libLLVM-16.0.6 (ORCJIT, apple-m2)
Threads: 8 default, 0 interactive, 4 GC (on 8 virtual cores)
Environment:
JULIA_NUM_THREADS = 8
JULIA_EDITOR = codeLoad packages:
using Pkg
Pkg.activate(pwd())
Pkg.instantiate()
Pkg.status() Activating project at `~/Documents/github.com/ucla-biostat-257/2025spring/slides/22-juliaopt`Status `~/Documents/github.com/ucla-biostat-257/2025spring/slides/22-juliaopt/Project.toml`
[1e616198] COSMO v0.8.9
[13f3f980] CairoMakie v0.13.5
[61c947e1] Clarabel v0.10.0
[f65535da] Convex v0.16.4
[31c24e10] Distributions v0.25.120
[5789e2e9] FileIO v1.17.0
⌅ [f6369f11] ForwardDiff v0.10.38
[2e9cd046] Gurobi v1.7.4
[b99e6be6] Hypatia v0.8.2
[916415d5] Images v0.26.2
[b6b21f68] Ipopt v1.10.3
[4076af6c] JuMP v1.25.0
[2ddba703] Juniper v0.9.3
[b8f27783] MathOptInterface v1.40.0
[1ec41992] MosekTools v0.15.9
[7f7a1694] Optimization v4.2.0
[fd9f6733] OptimizationMOI v0.5.3
[4e6fcdb7] OptimizationNLopt v0.3.2
[36348300] OptimizationOptimJL v0.4.3
[c946c3f1] SCS v2.1.0
[276daf66] SpecialFunctions v2.5.1
[10745b16] Statistics v1.11.1
[37e2e46d] LinearAlgebra v1.11.0
[9a3f8284] Random v1.11.0
[2f01184e] SparseArrays v1.11.0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`This lecture gives an overview of some optimization tools in Julia.
Statisticians do optimizations in daily life: maximum likelihood estimation, machine learning, …
Category of optimization problems:
Problems with analytical solutions: least squares, principle component analysis, canonical correlation analysis, …
Problems subject to Disciplined Convex Programming (DCP): linear programming (LP), quadratic programming (QP), second-order cone programming (SOCP), semidefinite programming (SDP), and geometric programming (GP).
Nonlinear programming (NLP): Newton type algorithms, Fisher scoring algorithm, EM algorithm, MM algorithms.
Large scale optimization: ADMM, SGD, …
Getting familiar with good optimization softwares broadens the scope and scale of problems we are able to solve in statistics. Following table lists some of the best optimization softwares.
| LP | MILP | SOCP | MISOCP | SDP | GP | NLP | MINLP | R | Matlab | Julia | Python | Cost | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| modeling tools | ||||||||||||||||
| cvx | x | x | x | x | x | x | x | x | x | A | ||||||
| Convex.jl | x | x | x | x | x | x | O | |||||||||
| Optimization.jl | x | x | x | x | x | x | x | O | ||||||||
| JuMP.jl | x | x | x | x | x | x | x | O | ||||||||
| MathOptInterface.jl | x | x | x | x | x | x | x | O | ||||||||
| convex solvers | ||||||||||||||||
| Mosek | x | x | x | x | x | x | x | x | x | x | x | A | ||||
| Gurobi | x | x | x | x | x | x | x | x | A | |||||||
| CPLEX | x | x | x | x | x | x | x | x | A | |||||||
| SCS | x | x | x | x | x | x | O | |||||||||
| COSMO.jl | x | x | x | x | O | |||||||||||
| Hypatia.jl (more cones) | x | x | x | x | O | |||||||||||
| NLP solvers | ||||||||||||||||
| NLopt | x | x | x | x | x | x | O | |||||||||
| Ipopt | x | x | x | x | x | x | O | |||||||||
| KNITRO | x | x | x | x | x | x | x | x | $ |
O: open source, A: free academic license, $: commercialFor more extended list of solvers, see here.
Difference between modeling tool and solvers
Modeling tools such as cvx (for Matlab) and Convex.jl (Julia analog of cvx) implement the disciplined convex programming (DCP) paradigm proposed by Grant and Boyd (2008) http://stanford.edu/~boyd/papers/disc_cvx_prog.html. DCP prescribes a set of simple rules from which users can construct convex optimization problems easily.
Solvers (Mosek, Gurobi, Cplex, SCS, COSMO, Hypatia, …) are concrete software implementation of optimization algorithms. My favorite ones are: Mosek/Gurobi/SCS for DCP and Ipopt/NLopt for nonlinear programming. Mosek and Gurobi are commercial software but free for academic use. SCS/Ipopt/NLopt are open source.
Modeling tools usually have the capability to use a variety of solvers. But modeling tools are solver agnostic so users do not have to worry about specific solver interface.
If you want to install the commercial solvers Gurobi or Mosek, instructions are below:
grbgetkey XXXXXXXXX command on terminal as suggested. It’ll retrieve a license file and put it under a specified folder, e.g., /Users/huazhou/Documents/Gurobi/gurobi.lic. 4. Set up the environmental variables. On my machine, I put following two lines in the ~/.julia/config/startup.jl file: ENV["GUROBI_HOME"] = "/Library/gurobi1201/macos_universal2" and ENV["GRB_LICENSE_FILE"] = "/Users/huazhou/Documents/Gurobi/gurobi.lic".~/mosek/.Standard convex problem classes like LP (linear programming), QP (quadratic programming), SOCP (second-order cone programming), SDP (semidefinite programming), and GP (geometric programming), are becoming a technology.
We illustrate optimization tools in Julia using microbiome analysis as an example.
16S microbiome sequencing techonology generates sequence counts of various organisms (OTUs, operational taxonomic units) in samples.
For statistical analysis, counts are normalized into proportions for each sample, resulting in a covariate matrix \(\mathbf{X}\) with all rows summing to 1. For identifiability, we need to add a sum-to-zero constraint to the regression cofficients. In other words, we need to solve a constrained least squares problem
\[
\text{minimize} \frac{1}{2} \|\mathbf{y} - \mathbf{X} \beta\|_2^2
\] subject to the constraint \(\sum_{j=1}^p \beta_j = 0\). For simplicity we ignore intercept and non-OTU covariates in this presentation.
Let’s first generate an artifical data set.
using Random, LinearAlgebra, SparseArrays
Random.seed!(257) # seed
n, p = 100, 50
X = rand(n, p)
# scale each row of X sum to 1
lmul!(Diagonal(1 ./ vec(sum(X, dims=2))), X)
# true β is a sparse vector with about 10% non-zero entries
β = sprandn(p, 0.1)
y = X * β + randn(n);The sum-to-zero contrained least squares is a standard quadratic programming (QP) problem so should be solved easily by any QP solver.
We use the Convex.jl package to model this QP problem. For a complete list of operations supported by Convex.jl, see https://jump.dev/Convex.jl/stable/operations/.
using Convex
β̂cls = Variable(size(X, 2))
problem = minimize(0.5sumsquares(y - X * β̂cls)) # objective
problem.constraints = [sum(β̂cls) == 0]; # constraint
problemProblem statistics
problem is DCP : true
number of variables : 1 (50 scalar elements)
number of constraints : 1 (1 scalar elements)
number of coefficients : 5_103
number of atoms : 7
Solution summary
termination status : OPTIMIZE_NOT_CALLED
primal status : NO_SOLUTION
dual status : NO_SOLUTION
Expression graph
minimize
└─ * (convex; positive)
├─ [0.5;;]
└─ qol (convex; positive)
├─ + (affine; real)
│ ├─ …
│ └─ …
└─ [1;;]
subject to
└─ == constraint (affine)
└─ + (affine; real)
├─ sum (affine; real)
│ └─ …
└─ [0;;]We first use the Mosek solver to solve this QP.
using MosekTools, MathOptInterface
const MOI = MathOptInterface
solver = MOI.OptimizerWithAttributes(Mosek.Optimizer, "LOG" => 1)
@time solve!(problem, solver)Problem
Name :
Objective sense : minimize
Type : CONIC (conic optimization problem)
Constraints : 1
Affine conic cons. : 1 (102 rows)
Disjunctive cons. : 0
Cones : 0
Scalar variables : 51
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - primal attempts : 1 successes : 1
Lin. dep. - dual attempts : 0 successes : 0
Lin. dep. - primal deps. : 0 dual deps. : 0
Presolve terminated. Time: 0.00
Optimizer - threads : 12
Optimizer - solved problem : the dual
Optimizer - Constraints : 50
Optimizer - Cones : 2
Optimizer - Scalar variables : 104 conic : 104
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.00
Factor - dense det. time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 1275 after factor : 1275
Factor - dense dim. : 0 flops : 3.03e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.9e+00 2.1e-01 1.7e+00 0.00e+00 3.535533906e-01 -3.535533906e-01 1.0e+00 0.00
1 1.3e-01 8.8e-03 2.6e-01 -9.42e-01 5.476668348e+00 1.694192885e+01 4.3e-02 0.00
2 1.9e-02 1.3e-03 3.4e-02 -2.35e-01 1.848385531e+01 2.813003558e+01 6.4e-03 0.00
3 7.7e-04 5.5e-05 4.1e-04 5.50e-01 2.814808382e+01 2.895785952e+01 2.6e-04 0.00
4 5.7e-05 4.0e-06 5.9e-06 9.72e-01 2.880931999e+01 2.884037712e+01 1.9e-05 0.00
5 4.5e-10 4.0e-12 3.1e-15 9.98e-01 2.884672314e+01 2.884672314e+01 5.7e-14 0.00
Optimizer terminated. Time: 0.00
0.011685 seconds (10.79 k allocations: 2.580 MiB, 55.17% compilation time: 100% of which was recompilation)[ Info: [Convex.jl] Compilation finished: 0.0 seconds, 2.256 MiB of memory allocatedProblem statistics
problem is DCP : true
number of variables : 1 (50 scalar elements)
number of constraints : 1 (1 scalar elements)
number of coefficients : 5_103
number of atoms : 7
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 28.8467
Expression graph
minimize
└─ * (convex; positive)
├─ [0.5;;]
└─ qol (convex; positive)
├─ + (affine; real)
│ ├─ …
│ └─ …
└─ [1;;]
subject to
└─ == constraint (affine)
└─ + (affine; real)
├─ sum (affine; real)
│ └─ …
└─ [0;;]# Check the status, optimal value, and minimizer of the problem
problem.status, problem.optval, β̂cls.value(MathOptInterface.OPTIMAL, 28.846723126126648, [5.50999308911286; 2.3835671892547334; … ; -3.5233708752162136; -14.863428523256783;;])# check constraint satisfication
sum(β̂cls.value)2.4868995751603507e-13Switch to Gurobi solver:
using Gurobi
solver = MOI.OptimizerWithAttributes(Gurobi.Optimizer, "OutputFlag" => 1)
@time solve!(problem, solver)Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter OutputFlag to value 1
Gurobi Optimizer version 12.0.2 build v12.0.2rc0 (mac64[arm] - Darwin 24.4.0 24E263)
CPU model: Apple M2 Max
Thread count: 12 physical cores, 12 logical processors, using up to 12 threads
Optimize a model with 103 rows, 153 columns and 5155 nonzeros
Model fingerprint: 0xceaf534d
Model has 1 quadratic constraint
Coefficient statistics:
Matrix range [3e-05, 1e+00]
QMatrix range [1e+00, 1e+00]
Objective range [5e-01, 5e-01]
Bounds range [0e+00, 0e+00]
RHS range [3e-02, 3e+00]
Presolve removed 1 rows and 1 columns
Presolve time: 0.00s
Presolved: 102 rows, 152 columns, 5152 nonzeros
Presolved model has 1 second-order cone constraint
Ordering time: 0.00s
Barrier statistics:
Free vars : 50
AA' NZ : 5.150e+03
Factor NZ : 5.253e+03
Factor Ops : 3.590e+05 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 7.13616591e+00 0.00000000e+00 1.43e+01 1.39e-02 7.28e-02 0s
1 3.01025068e+00 6.64661859e-01 5.49e+00 2.91e-05 2.46e-02 0s
2 4.03572275e+00 3.01841471e+00 3.28e+00 3.53e-06 2.31e-02 0s
3 1.37424301e+01 9.41379966e+00 6.97e-01 9.61e-07 5.61e-02 0s
4 2.24325118e+01 2.47075686e+01 2.01e-01 3.44e-06 2.72e-02 0s
5 3.23085168e+01 1.89965179e+01 2.21e-07 2.88e-05 1.31e-01 0s
6 2.97349069e+01 2.87096350e+01 9.90e-10 4.39e-06 1.00e-02 0s
7 2.88512338e+01 2.88129938e+01 4.53e-11 2.26e-07 3.75e-04 0s
8 2.88475708e+01 2.88462464e+01 2.26e-11 2.71e-08 1.30e-05 0s
9 2.88467247e+01 2.88467190e+01 1.07e-11 3.92e-09 6.30e-08 0s
Barrier solved model in 9 iterations and 0.00 seconds (0.01 work units)
Optimal objective 2.88467247e+01
User-callback calls 74, time in user-callback 0.00 sec
0.011327 seconds (101.53 k allocations: 6.667 MiB)[ Info: [Convex.jl] Compilation finished: 0.01 seconds, 6.649 MiB of memory allocatedProblem statistics
problem is DCP : true
number of variables : 1 (50 scalar elements)
number of constraints : 1 (1 scalar elements)
number of coefficients : 5_103
number of atoms : 7
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : NO_SOLUTION
objective value : 28.8467
Expression graph
minimize
└─ * (convex; positive)
├─ [0.5;;]
└─ qol (convex; positive)
├─ + (affine; real)
│ ├─ …
│ └─ …
└─ [1;;]
subject to
└─ == constraint (affine)
└─ + (affine; real)
├─ sum (affine; real)
│ └─ …
└─ [0;;]# Check the status, optimal value, and minimizer of the problem
problem.status, problem.optval, β̂cls.value(MathOptInterface.OPTIMAL, 28.84672474228832, [5.510457528212885; 2.3837733033643653; … ; -3.523669336789277; -14.864701503734674;;])# check constraint satisfication
sum(β̂cls.value)-1.0231815394945443e-12Switch to COSMO solver (pure Julia implementation):
# Use COSMO solver
using COSMO
solver = MOI.OptimizerWithAttributes(COSMO.Optimizer, "max_iter" => 50_000, "verbose" => false)
@time solve!(problem, solver) 0.319794 seconds (531.85 k allocations: 110.897 MiB, 4.14% gc time, 10.95% compilation time: 100% of which was recompilation)[ Info: [Convex.jl] Compilation finished: 0.03 seconds, 3.366 MiB of memory allocatedProblem statistics
problem is DCP : true
number of variables : 1 (50 scalar elements)
number of constraints : 1 (1 scalar elements)
number of coefficients : 5_103
number of atoms : 7
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 28.8809
Expression graph
minimize
└─ * (convex; positive)
├─ [0.5;;]
└─ qol (convex; positive)
├─ + (affine; real)
│ ├─ …
│ └─ …
└─ [1;;]
subject to
└─ == constraint (affine)
└─ + (affine; real)
├─ sum (affine; real)
│ └─ …
└─ [0;;]# Check the status, optimal value, and minimizer of the problem
problem.status, problem.optval, β̂cls.value(MathOptInterface.OPTIMAL, 28.880882002894065, [5.509993060139416; 2.383567050955549; … ; -3.523370764002651; -14.863428540284978;;])We see COSMO have a looser criterion for constraint satisfication, resulting a lower objective value.
sum(β̂cls.value)2.444553715008624e-7Switch to the open source SCS solver:
# Use SCS solver
using SCS
solver = MOI.OptimizerWithAttributes(SCS.Optimizer, "verbose" => 5000)
@time solve!(problem, solver)------------------------------------------------------------------
SCS v3.2.7 - Splitting Conic Solver
(c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem: variables n: 51, constraints m: 103
cones: z: primal zero / dual free vars: 1
q: soc vars: 102, qsize: 1
settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
max_iters: 100000, normalize: 1, rho_x: 1.00e-06
acceleration_lookback: 10, acceleration_interval: 10
compiled with openmp parallelization enabled
lin-sys: sparse-direct-amd-qdldl
nnz(A): 5052, nnz(P): 0
------------------------------------------------------------------
iter | pri res | dua res | gap | obj | scale | time (s)
------------------------------------------------------------------
0| 8.17e+00 3.65e+00 3.88e+01 1.86e+01 1.00e-01 4.70e-04
250| 1.53e-03 1.61e-04 8.02e-02 2.88e+01 4.93e-03 4.05e-03
325| 8.77e-08 2.27e-08 2.23e-09 2.88e+01 4.93e-03 5.03e-03
------------------------------------------------------------------
status: solved
timings: total: 5.03e-03s = setup: 3.46e-04s + solve: 4.69e-03s
lin-sys: 3.40e-03s, cones: 7.41e-05s, accel: 3.41e-05s
------------------------------------------------------------------
objective = 28.846724
------------------------------------------------------------------
0.012632 seconds (9.55 k allocations: 2.981 MiB, 35.70% compilation time: 100% of which was recompilation)[ Info: [Convex.jl] Compilation finished: 0.0 seconds, 2.394 MiB of memory allocatedProblem statistics
problem is DCP : true
number of variables : 1 (50 scalar elements)
number of constraints : 1 (1 scalar elements)
number of coefficients : 5_103
number of atoms : 7
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 28.8467
Expression graph
minimize
└─ * (convex; positive)
├─ [0.5;;]
└─ qol (convex; positive)
├─ + (affine; real)
│ ├─ …
│ └─ …
└─ [1;;]
subject to
└─ == constraint (affine)
└─ + (affine; real)
├─ sum (affine; real)
│ └─ …
└─ [0;;]# Check the status, optimal value, and minimizer of the problem
problem.status, problem.optval, β̂cls.value(MathOptInterface.OPTIMAL, 28.846724444532505, [5.509993089050771; 2.3835671885418215; … ; -3.523370874675494; -14.863428523247924;;])# check constraint satisfication
sum(β̂cls.value)1.3278267374516872e-9Switch to the open source Hypatia (pure Julia) solver:
# Use Hypatia solver
using Hypatia
solver = MOI.OptimizerWithAttributes(Hypatia.Optimizer, "verbose" => 1)
@time solve!(problem, solver)
iter p_obj d_obj | abs_gap x_feas z_feas | tau kap mu | dir_res prox step alpha
0 8.5628e-01 -2.9196e-01 | 2.00e+00 5.59e-02 5.59e-01 | 1.00e+00 1.00e+00 1.00e+00 |
1 4.3997e+00 7.8189e+00 | 1.45e-01 3.04e-02 3.04e-01 | 1.84e-01 8.43e-01 1.00e-01 | 5.3e-15 5.5e-01 co-a 9.00e-01
2 1.4279e+01 2.2121e+01 | 1.07e-02 1.19e-02 1.19e-01 | 4.70e-02 3.90e-01 9.67e-03 | 8.9e-16 8.9e-01 co-a 9.00e-01
3 2.3867e+01 2.7924e+01 | 1.80e-03 3.59e-03 3.59e-02 | 2.34e-02 9.81e-02 1.36e-03 | 3.9e-16 6.8e-01 co-a 8.50e-01
4 2.8645e+01 2.8834e+01 | 4.44e-05 1.31e-04 1.31e-03 | 1.92e-02 3.73e-03 3.86e-05 | 1.6e-15 8.5e-01 co-a 9.70e-01
5 2.8845e+01 2.8846e+01 | 8.85e-07 1.29e-06 1.29e-05 | 1.96e-02 1.53e-05 3.95e-07 | 4.9e-16 3.3e-01 co-a 9.90e-01
6 2.8847e+01 2.8847e+01 | 8.17e-09 1.42e-08 1.42e-07 | 1.77e-02 1.44e-07 3.57e-09 | 9.7e-13 6.5e-01 co-a 9.90e-01
7 2.8847e+01 2.8847e+01 | 5.97e-12 1.14e-11 1.70e-10 | 1.48e-02 1.99e-10 2.97e-12 | 3.8e-11 1.8e-02 co-a 9.99e-01
optimal solution found; terminating
status is Optimal after 7 iterations and 0.003 seconds
0.014444 seconds (16.92 k allocations: 3.808 MiB, 62.27% compilation time: 100% of which was recompilation)[ Info: [Convex.jl] Compilation finished: 0.0 seconds, 1.732 MiB of memory allocatedProblem statistics
problem is DCP : true
number of variables : 1 (50 scalar elements)
number of constraints : 1 (1 scalar elements)
number of coefficients : 5_103
number of atoms : 7
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 28.8467
Expression graph
minimize
└─ * (convex; positive)
├─ [0.5;;]
└─ qol (convex; positive)
├─ + (affine; real)
│ ├─ …
│ └─ …
└─ [1;;]
subject to
└─ == constraint (affine)
└─ + (affine; real)
├─ sum (affine; real)
│ └─ …
└─ [0;;]# Check the status, optimal value, and minimizer of the problem
problem.status, problem.optval, β̂cls.value(MathOptInterface.OPTIMAL, 28.84672311024055, [5.509993088754246; 2.3835671898357353; … ; -3.523370874921378; -14.863428523366206;;])# check constraint satisfication
sum(β̂cls.value)0.0Switch to the open source Clarabel (pure Julia) solver:
# Use Hypatia solver
using Clarabel
solver = MOI.OptimizerWithAttributes(Clarabel.Optimizer)
@time solve!(problem, solver)[ Info: [Convex.jl] Compilation finished: 0.02 seconds, 3.069 MiB of memory allocated-------------------------------------------------------------
Clarabel.jl v0.10.0 - Clever Acronym
(c) Paul Goulart
University of Oxford, 2022
-------------------------------------------------------------
problem:
variables = 51
constraints = 103
nnz(P) = 0
nnz(A) = 5052
cones (total) = 2
: Zero = 1, numel = 1
: SecondOrder = 1, numel = 102
settings:
linear algebra: direct / qdldl, precision: Float64
max iter = 200, time limit = Inf, max step = 0.990
tol_feas = 1.0e-08, tol_gap_abs = 1.0e-08, tol_gap_rel = 1.0e-08,
static reg : on, ϵ1 = 1.0e-08, ϵ2 = 4.9e-32
dynamic reg: on, ϵ = 1.0e-13, δ = 2.0e-07
iter refine: on, reltol = 1.0e-13, abstol = 1.0e-12,
max iter = 10, stop ratio = 5.0
equilibrate: on, min_scale = 1.0e-04, max_scale = 1.0e+04
max iter = 10
iter pcost dcost gap pres dres k/t μ step
---------------------------------------------------------------------------------------------
0 3.0999e-18 -8.7522e+00 8.75e+00 8.87e-02 1.86e-01 1.00e+00 9.64e+01 ------
1 4.0721e+00 1.5813e+02 3.78e+01 1.40e-01 5.45e-02 1.72e+02 1.15e+01 9.90e-01
2 2.9150e+01 3.5232e+02 1.11e+01 2.12e-02 4.38e-03 3.28e+02 8.50e-01 9.26e-01
3 4.7036e+01 4.7082e+01 9.89e-04 4.28e-03 4.14e-03 1.27e+00 6.15e-01 4.12e-01
4 3.6893e+01 4.1455e+01 1.24e-01 5.81e-04 1.13e-03 4.70e+00 1.38e-01 8.15e-01
5 2.8671e+01 2.9210e+01 1.88e-02 4.33e-05 9.52e-05 5.47e-01 1.19e-02 9.28e-01
6 2.8857e+01 2.8874e+01 5.95e-04 2.10e-06 5.14e-06 1.76e-02 6.13e-04 9.90e-01
7 2.8847e+01 2.8847e+01 7.55e-06 2.59e-08 6.34e-08 2.23e-04 7.56e-06 9.88e-01
8 2.8847e+01 2.8847e+01 1.79e-07 6.11e-10 1.49e-09 5.28e-06 1.78e-07 9.76e-01
9 2.8847e+01 2.8847e+01 6.97e-09 1.81e-11 5.75e-11 2.06e-07 6.86e-09 9.62e-01
---------------------------------------------------------------------------------------------
Terminated with status = solved
solve time = 4.91ms
0.032203 seconds (31.45 k allocations: 5.747 MiB, 72.66% compilation time: 100% of which was recompilation)Problem statistics
problem is DCP : true
number of variables : 1 (50 scalar elements)
number of constraints : 1 (1 scalar elements)
number of coefficients : 5_103
number of atoms : 7
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 28.8467
Expression graph
minimize
└─ * (convex; positive)
├─ [0.5;;]
└─ qol (convex; positive)
├─ + (affine; real)
│ ├─ …
│ └─ …
└─ [1;;]
subject to
└─ == constraint (affine)
└─ + (affine; real)
├─ sum (affine; real)
│ └─ …
└─ [0;;]# Check the status, optimal value, and minimizer of the problem
problem.status, problem.optval, β̂cls.value(MathOptInterface.OPTIMAL, 28.846723012739073, [5.509993089173661; 2.3835671892605546; … ; -3.52337087522751; -14.86342852323002;;])# check constraint satisfication
sum(β̂cls.value)-7.105427357601002e-15Suppose we want to know which organisms (OTU) are associated with the response. We can answer this question using a sum-to-zero contrained lasso \[ \text{minimize} \frac 12 \|\mathbf{y} - \mathbf{X} \beta\|_2^2 + \lambda \|\beta\|_1 \] subject to the constraint \(\sum_{j=1}^p \beta_j = 0\). Varying \(\lambda\) from small to large values will generate a solution path.
using Convex
# # Use Mosek solver
# using Mosek
# solver = Mosek.Optimizer
# # Use Gurobi solver
# using Gurobi
# solver = Gurobi.Optimizer
# # Use SCS solver
# using SCS
# solver = SCS.Optimizer
# Use Hypatia solver
using Hypatia
solver = Hypatia.Optimizer
# # Use Clarabel solver
# using Clarabel
# solver = Clarabel.Optimizer
# solve at a grid of λ
λgrid = 0:0.01:0.35
# holder for solution path
β̂path = zeros(length(λgrid), size(X, 2)) # each row is β̂ at a λ
# optimization variable
β̂classo = Variable(size(X, 2))
# obtain solution path using warm start
@time for i in 1:length(λgrid)
λ = λgrid[i]
# define optimization problem
# objective
problem = minimize(0.5sumsquares(y - X * β̂classo) + λ * sum(abs, β̂classo))
# constraint
problem.constraints += sum(β̂classo) == 0 # constraint
solve!(problem, solver; silent = true)
β̂path[i, :] = β̂classo.value
end 0.516054 seconds (1.37 M allocations: 219.045 MiB, 8.30% gc time, 7.80% compilation time: 100% of which was recompilation)using CairoMakie
f = Figure()
Axis(
f[1, 1],
title = "Sum-to-Zero Lasso",
xlabel = L"\lambda",
ylabel = L"\beta_j"
)
series!(λgrid, β̂path', color = :glasbey_category10_n256)
f
Suppose we want to do variable selection not at the OTU level, but at the Phylum level. OTUs are clustered into various Phyla. We can answer this question using a sum-to-zero contrained group lasso \[ \text{minimize} \frac 12 \|\mathbf{y} - \mathbf{X} \beta\|_2^2 + \lambda \sum_j \|\mathbf{\beta}_j\|_2 \] subject to the constraint \(\sum_{j=1}^p \beta_j = 0\), where \(\mathbf{\beta}_j\) are regression coefficients corresponding to the \(j\)-th phylum. This is a second-order cone programming (SOCP) problem readily modeled by Convex.jl.
Let’s assume each 10 contiguous OTUs belong to one Phylum.
# # Use Mosek solver
# using Mosek
# solver = Mosek.Optimizer
# # Use Gurobi solver
# using Gurobi
# solver = Gurobi.Optimizer
# # Use SCS solver
# using SCS
# solver = SCS.Optimizer
# Use Hypatia solver
using Hypatia
solver = Hypatia.Optimizer
# # Use Clarabel solver
# using Clarabel
# solver = Clarabel.Optimizer
# solve at a grid of λ
λgrid = 0.0:0.005:0.5
β̂pathgrp = zeros(length(λgrid), size(X, 2)) # each row is β̂ at a λ
β̂classo = Variable(size(X, 2))
@time for i in 1:length(λgrid)
λ = λgrid[i]
# loss
obj = 0.5sumsquares(y - X * β̂classo)
# group lasso penalty term
for j in 1:(size(X, 2) ÷ 10)
βj = β̂classo[(10(j - 1) + 1) : 10j]
obj = obj + λ * norm(βj)
end
problem = minimize(obj)
# constraint
problem.constraints = [sum(β̂classo) == 0] # constraint
solve!(problem, solver; silent = true)
β̂pathgrp[i, :] = β̂classo.value
end 0.618763 seconds (2.12 M allocations: 438.603 MiB, 12.08% gc time)We see it took Mosek <2 second to solve this seemingly hard optimization problem at 101 different \(\lambda\) values.
using CairoMakie
f = Figure()
Axis(
f[1, 1],
title = "Sum-to-Zero Group Lasso",
xlabel = L"\lambda",
ylabel = L"\beta_j"
)
series!(λgrid, β̂pathgrp', color = :glasbey_category10_n256)
f
Load the \(128 \times 128\) Lena picture with missing pixels.
using FileIO
lena = load("lena128missing.png")
# convert to real matrices
Y = Float64.(lena)128×128 Matrix{Float64}:
0.0 0.0 0.635294 0.0 … 0.0 0.0 0.627451
0.627451 0.623529 0.0 0.611765 0.0 0.0 0.388235
0.611765 0.611765 0.0 0.0 0.403922 0.219608 0.0
0.0 0.0 0.611765 0.0 0.223529 0.176471 0.192157
0.611765 0.0 0.615686 0.615686 0.0 0.0 0.0
0.0 0.0 0.0 0.619608 … 0.0 0.0 0.2
0.607843 0.0 0.623529 0.0 0.176471 0.192157 0.0
0.0 0.0 0.623529 0.0 0.0 0.0 0.215686
0.619608 0.619608 0.0 0.0 0.2 0.0 0.207843
0.0 0.0 0.635294 0.635294 0.2 0.192157 0.188235
0.635294 0.0 0.0 0.0 … 0.192157 0.180392 0.0
0.631373 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.627451 0.635294 0.666667 0.172549 0.0 0.184314
⋮ ⋱ ⋮
0.0 0.129412 0.0 0.541176 0.0 0.286275 0.0
0.14902 0.129412 0.196078 0.537255 0.345098 0.0 0.0
0.215686 0.0 0.262745 0.0 0.301961 0.0 0.207843
0.345098 0.341176 0.356863 0.513725 0.0 0.0 0.231373
0.0 0.0 0.0 0.0 … 0.0 0.243137 0.258824
0.298039 0.415686 0.458824 0.0 0.0 0.0 0.258824
0.0 0.368627 0.4 0.0 0.0 0.0 0.235294
0.0 0.0 0.34902 0.0 0.0 0.239216 0.207843
0.219608 0.0 0.0 0.0 0.0 0.0 0.2
0.0 0.219608 0.235294 0.356863 … 0.0 0.0 0.0
0.196078 0.207843 0.211765 0.0 0.0 0.270588 0.345098
0.192157 0.0 0.196078 0.309804 0.266667 0.356863 0.0We fill out the missin pixels uisng a matrix completion technique developed by Candes and Tao \[ \text{minimize } \|\mathbf{X}\|_* \] \[ \text{subject to } x_{ij} = y_{ij} \text{ for all observed entries } (i, j). \] Here \(\|\mathbf{M}\|_* = \sum_i \sigma_i(\mathbf{M})\) is the nuclear norm. In words we seek the matrix with minimal nuclear norm that agrees with the observed entries. This is a semidefinite programming (SDP) problem readily modeled by Convex.jl.
This example takes longer because of high dimensionality. COSMO.jl seems to be the fastest solver for this problem. Other solvers take excessively long time.
# Use COSMO solver (fast)
using COSMO
solver = COSMO.Optimizer
# # Use Hypatia solver (slow)
# using Hypatia
# solver = Hypatia.Optimizer()
# # Use Clarabel solver (slow)
# using Clarabel
# solver = Clarabel.Optimizer()
# Linear indices of obs. entries
obsidx = findall(Y[:] .≠ 0.0)
# Create optimization variables
X = Variable(size(Y))
# Set up optmization problem
problem = minimize(nuclearnorm(X))
problem.constraints += X[obsidx] == Y[obsidx]
# Solve the problem by calling solve
@time solve!(problem, solver)[ Info: [Convex.jl] Compilation finished: 0.23 seconds, 193.573 MiB of memory allocated------------------------------------------------------------------
COSMO v0.8.9 - A Quadratic Objective Conic Solver
Michael Garstka
University of Oxford, 2017 - 2022
------------------------------------------------------------------
Problem: x ∈ R^{32897},
constraints: A ∈ R^{41025x32897} (41281 nnz),
matrix size to factor: 73922x73922,
Floating-point precision: Float64
Sets: DensePsdConeTriangle of dim: 32896 (256x256)
ZeroSet of dim: 8128
Nonnegatives of dim: 1
Settings: ϵ_abs = 1.0e-05, ϵ_rel = 1.0e-05,
ϵ_prim_inf = 1.0e-04, ϵ_dual_inf = 1.0e-04,
ρ = 0.1, σ = 1e-06, α = 1.6,
max_iter = 5000,
scaling iter = 10 (on),
check termination every 25 iter,
check infeasibility every 40 iter,
KKT system solver: QDLDL
Acc: Anderson Type2{QRDecomp},
Memory size = 15, RestartedMemory,
Safeguarded: true, tol: 2.0
Setup Time: 12945.24ms
Iter: Objective: Primal Res: Dual Res: Rho:
1 -1.4585e+03 1.5985e+01 5.9854e-01 1.0000e-01
25 1.4525e+02 5.1105e-02 1.1356e-03 1.0000e-01
50 1.4758e+02 1.1725e-02 1.4834e-03 6.8658e-01
75 1.4797e+02 5.5160e-04 4.7489e-05 6.8658e-01
100 1.4797e+02 1.7304e-05 1.3870e-06 6.8658e-01
------------------------------------------------------------------
>>> Results
Status: Solved
Iterations: 100
Optimal objective: 148
Runtime: 13.625s (13625.11ms)
13.862155 seconds (510.42 k allocations: 512.114 MiB, 0.34% gc time, 1.38% compilation time: 100% of which was recompilation)Problem statistics
problem is DCP : true
number of variables : 1 (16_384 scalar elements)
number of constraints : 1 (8_128 scalar elements)
number of coefficients : 8_128
number of atoms : 3
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 147.9711
Expression graph
minimize
└─ nuclearnorm (convex; positive)
└─ 128×128 real variable (id: 420…982)
subject to
└─ == constraint (affine)
└─ + (affine; real)
├─ index (affine; real)
│ └─ …
└─ 8128×1 Matrix{Float64}using Images
# Result by nuclear norm minimization
colorview(Gray, X.value)
We can also complete the matrix by the nonnegative matrix factorization (NMF) approach. The NMF problem is a non-convex optimization problem. \[ \text{minimize} \|P_{\Omega}(\mathbf{Y}) - P_{\Omega}(\mathbf{V}\mathbf{W})\|_F^2 \] \[ \text{subject to } \mathbf{V} \geq 0, \mathbf{W} \geq 0 \] where \(P_{\Omega}(\mathbf{Y})\) is the projection of \(\mathbf{Y}\) onto the observed entries \(\Omega\). The NMF problem is a non-convex optimization problem because it has multiple local optima. However, it can be solved using alternating least squares (ALS) or multiplicative update rules. The ALS algorithm is a popular method for solving the NMF problem. It alternates between fixing one variable and solving for the other, and vice versa.
function mc_by_nnmf(Y, r; solver = SCS.Optimizer)
# Linear indices of obs. entries
obsidx = findall(Y[:] .≠ 0.0)
yobs = Y[obsidx]
# Create optimization variables
V = Variable(size(Y, 1), r)
W = Variable(r, size(Y, 2))
# Set up optmization problem
problem = minimize(norm(yobs - (V * W)[obsidx]), [V >= 0, W >= 0])
# Initialize value of W
set_value!(W, rand(r, size(Y, 2)))
# We'll do 10 iterations of alternating minimization
for iter in 1:20
# Solve for V, with W fixed
fix!(W)
solve!(problem, solver; warmstart = iter > 1 ? true : false, silent = true)
free!(W)
# Solve for W, with V fixed
fix!(V)
solve!(problem, solver; warmstart = true, silent = true)
free!(V)
end
# Return the solution
return V.value, W.value
endmc_by_nnmf (generic function with 1 method)V, W = mc_by_nnmf(Y, 30; solver = Gurobi.Optimizer)Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
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Set parameter Username
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Set parameter Username
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Set parameter Username
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Set parameter Username
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Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
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Set parameter Username
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Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
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Set parameter Username
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Set parameter Username
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Set parameter Username
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Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
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Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15
Set parameter Username
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Set parameter Username
Academic license - for non-commercial use only - expires 2026-02-15([0.03558936365695675 0.09963359790567292 … 0.001807424224772325 2.8542616835645243e-11; 0.01885998122844864 0.12714463761475492 … 0.00562394905294704 3.358410961702319e-10; … ; 0.050942574505134605 0.006123770500843592 … 0.10524562898251237 0.08774136866099994; 0.07223482340443767 0.010225933560439444 … 0.11667564009521098 0.09297504327476738], [0.6636404981012094 0.2169827368432252 … 0.3268122809244541 1.9189964304527554; 0.583010787675328 1.2629974910345814 … 9.711047381150691e-10 0.06543190569928403; … ; 0.44883297716563164 0.43641935398670023 … 1.5998200137167535e-9 9.49988388078683e-8; 6.226170617867564e-10 4.2629100916064024e-9 … 0.14559006387260917 0.40718501213557473])# Result by NNMF
colorview(Gray, V * W)
We use MLE of Gamma distribution to illustrate some rudiments of nonlinear programming (NLP) in Julia.
Let \(x_1,\ldots,x_m\) be a random sample from the gamma density \[ f(x) = \Gamma(\alpha)^{-1} \beta^{\alpha} x^{\alpha-1} e^{-\beta x} \] on \((0,\infty)\). The loglikelihood function is \[ L(\alpha, \beta) = m [- \ln \Gamma(\alpha) + \alpha \ln \beta + (\alpha - 1)\overline{\ln x} - \beta \bar x], \] where \(\overline{x} = \frac{1}{m} \sum_{i=1}^m x_i\) and \(\overline{\ln x} = \frac{1}{m} \sum_{i=1}^m \ln x_i\).
using Distributions, Optimization, Random, Statistics, SpecialFunctions
Random.seed!(257)
# θ = (α, β)
function gamma_logpdf(x::Vector{<:Real}, θ::Vector{<:Real})
m = length(x)
avg = mean(x)
α, β = θ[1], θ[2]
logavg = sum(log, x) / m
m * (- log(gamma(α)) + α * log(β) + (α - 1) * logavg - β * avg)
end
# test data
x = rand(5)
gamma_logpdf(x, [1.0, 1.0])WARNING: using Distributions.Categorical in module Main conflicts with an existing identifier.
WARNING: using Distributions.scale! in module Main conflicts with an existing identifier.-3.3064167958026847Generate data.
Random.seed!(257)
(n, p) = (1000, 2)
(α, β) = 5rand(p)
x = rand(Gamma(α, β), n)
println("True parameter values:")
println("α = ", α, ", β = ", β)True parameter values:
α = 2.838860122889505, β = 4.699007334188252We use Optimization.jl to define and solve our NLP problem.
using ForwardDiff, Optimization
# loss function and gradient (by auto-diff)
loss = (θ, p) -> -gamma_logpdf(x, θ)
# more auto-diff choices: https://docs.sciml.ai/Optimization/stable/API/ad/
optf = OptimizationFunction(loss, Optimization.AutoForwardDiff())
# start point
θ₀ = [1.0, 1.0]
# optimization problem
prob = OptimizationProblem(optf, θ₀, lb = [0.0, 0.0], ub = [Inf, Inf])OptimizationProblem. In-place: true u0: 2-element Vector{Float64}: 1.0 1.0
List of algorithms in Optim.jl.
using OptimizationOptimJL
# BFGS algorithm (gradient based)
sol = solve(prob, BFGS())retcode: Success
u: 2-element Vector{Float64}:
2.9457402468066
0.2170674128195337# Nelder-Mead algorithm (gradient free)
sol = solve(prob, NelderMead())retcode: Success
u: 2-element Vector{Float64}:
2.9457349343640113
0.217067158037371# Conjugate Gradient algorithm (gradient based)
sol = solve(prob, ConjugateGradient())retcode: Success
u: 2-element Vector{Float64}:
2.9457402467986586
0.21706741281893602List of algorithms in OptimizationNLopt.jl.
using OptimizationNLopt
sol = solve(prob, NLopt.LD_LBFGS())retcode: Success
u: 2-element Vector{Float64}:
2.945740246710059
0.21706741281080016List of algorithms in OptimizationMOI.jl.
using OptimizationMOI, Ipopt
# Ipopt
sol = solve(prob, Ipopt.Optimizer())
******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
Ipopt is released as open source code under the Eclipse Public License (EPL).
For more information visit https://github.com/coin-or/Ipopt
******************************************************************************
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 2
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.3570624e+04 0.00e+00 9.90e+01 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 4.2328683e+03 0.00e+00 3.32e+02 -1.0 1.93e+01 - 1.00e+00 5.12e-02f 1
2 4.0086073e+03 0.00e+00 1.51e+02 -1.0 8.38e-02 - 7.22e-01 1.00e+00f 1
3 3.7647589e+03 0.00e+00 7.75e+01 -1.0 2.59e-01 - 7.47e-01 1.00e+00f 1
4 3.5568603e+03 0.00e+00 3.56e+01 -1.0 4.46e-01 - 8.02e-01 1.00e+00f 1
5 3.4205063e+03 0.00e+00 1.56e+01 -1.0 6.28e-01 - 1.00e+00 1.00e+00f 1
6 3.3700715e+03 0.00e+00 4.48e+00 -1.0 6.41e-01 - 1.00e+00 1.00e+00f 1
7 3.3636757e+03 0.00e+00 8.00e-01 -1.0 4.07e-01 - 1.00e+00 1.00e+00f 1
8 3.3636188e+03 0.00e+00 1.22e-03 -1.7 2.01e-02 - 1.00e+00 1.00e+00f 1
9 3.3635946e+03 0.00e+00 9.18e-04 -3.8 2.68e-02 - 1.00e+00 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 3.3635946e+03 0.00e+00 4.86e-08 -5.7 1.94e-04 - 1.00e+00 1.00e+00f 1
11 3.3635946e+03 0.00e+00 7.90e-12 -8.6 2.47e-06 - 1.00e+00 1.00e+00f 1
Number of Iterations....: 11
(scaled) (unscaled)
Objective...............: 2.6757578428056711e+01 3.3635945815883747e+03
Dual infeasibility......: 7.9016926883909898e-12 9.9329207923314562e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5076199894927938e-09 3.1522348078017871e-07
Overall NLP error.......: 2.5076199894927938e-09 3.1522348078017871e-07
Number of objective function evaluations = 12
Number of objective gradient evaluations = 12
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 11
Total seconds in IPOPT = 0.650
EXIT: Optimal Solution Found.retcode: Success
u: 2-element Vector{Float64}:
2.945740250057661
0.2170674130823058